| L(s) = 1 | + 0.298·2-s + 0.0958·3-s − 1.91·4-s + 0.959·5-s + 0.0285·6-s + 1.32·7-s − 1.16·8-s − 2.99·9-s + 0.286·10-s − 2.70·11-s − 0.183·12-s − 2.46·13-s + 0.396·14-s + 0.0919·15-s + 3.47·16-s + 17-s − 0.891·18-s − 2.75·19-s − 1.83·20-s + 0.127·21-s − 0.805·22-s − 1.10·23-s − 0.111·24-s − 4.07·25-s − 0.734·26-s − 0.574·27-s − 2.54·28-s + ⋯ |
| L(s) = 1 | + 0.210·2-s + 0.0553·3-s − 0.955·4-s + 0.429·5-s + 0.0116·6-s + 0.502·7-s − 0.412·8-s − 0.996·9-s + 0.0905·10-s − 0.814·11-s − 0.0528·12-s − 0.683·13-s + 0.105·14-s + 0.0237·15-s + 0.868·16-s + 0.242·17-s − 0.210·18-s − 0.631·19-s − 0.410·20-s + 0.0278·21-s − 0.171·22-s − 0.231·23-s − 0.0228·24-s − 0.815·25-s − 0.144·26-s − 0.110·27-s − 0.480·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 0.298T + 2T^{2} \) |
| 3 | \( 1 - 0.0958T + 3T^{2} \) |
| 5 | \( 1 - 0.959T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 7.14T + 29T^{2} \) |
| 31 | \( 1 + 3.87T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 0.411T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 6.68T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 + 8.72T + 89T^{2} \) |
| 97 | \( 1 - 1.38T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.788096977624038668881315416756, −9.198083588718381340540887960335, −8.200711032294994984403702160888, −7.65053325198669614587956527945, −6.06119166093719336825092509959, −5.41338280470408583454848280296, −4.59306549793070511257116034085, −3.36995157199993130740470841966, −2.12025008659549154538232489176, 0,
2.12025008659549154538232489176, 3.36995157199993130740470841966, 4.59306549793070511257116034085, 5.41338280470408583454848280296, 6.06119166093719336825092509959, 7.65053325198669614587956527945, 8.200711032294994984403702160888, 9.198083588718381340540887960335, 9.788096977624038668881315416756
